Statistical Physics, Second Revised and Enlarged Edition

Entropyand disorder 113

exactlyparallelsthederivation of(10.2), the resultbeing

S=−NkkkB

∑

j

PPjlnPPj (10.3)

wherePPj =nnj/N(strictlynn∗j/N)isdefinedtobethefraction oftheNparticles
inthe statej. This important equation can be taken as an alternative statisti-
cal definition of entropy, and indeed it can be applied with greater generality
than (1.5).

1 0.1.3 Gases

The same approach,ofexpressingSdirectly in terms ofthedistribution, canbe
adopted forgases, usingthe expressions fort∗derived in Chapter 5 .Again usingStir-
ling’s approximation together with a little rearrangement, one obtains the following
results:
For FDgases, from ( 5 .4),

S=kkkB

∑

i

gi[−−fffilnfffi−( 1 −fffi)ln( 1 −fffi)] (10.4)

For BEgases, from (5.5),

S=kkkB

∑

i

gi[−−fffilnfffi+( 1 +fffi)ln( 1 +fffi)] (10.5)

For MB gases, from (5.6), or from the dilute (((fffi 1 )limit of(10.4)or(10.5),

S=kkkB

∑

i

gi[−−fffilnfffi]+NkkkB (10.6)

In these equations, the sums are over allgroupsiof states; however, another way
of writing

∑

igiis simply as a sum over all states. Therefore there is much simi-
laritybetween thefirst (common) term ofthese expressions and(10.3)forlocalized
particles. We maywrite (10.3) as

S=kkkB

∑

j

[−nnjlnnnj]+NkkkBlnN

Recognizing thatnnjandfffiarebothdefinedas the number ofparticles per state
(thefillingfactor), thesimilarityis explicit. Infact theonly differencebetween
equation (10.3) for localized particles and (10.6) for MB gas particles is an addition
ofkkkBlnN!to the entropy, as we shouldexpectfrom thedifferent extensivity ofthe
two cases (section 6 .3.3).